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VERTICAL INTEGRATION IN 
COMPETITIVE MARKETS UNDER UNCERTAINTY 

By 



Number 174 



Dennis W. Carlton 



*^^ 



April 1976 




massachusetts 
I institute of 

technology 

50 memorial drive 
Cambridge, mass. 02139 



VERTICAL INTEGRATION IN 

COMPETITIVE MARKETS UNDER UNCERTAINTY x 

By 
Dennis W. Carlton 
Number 174 ' April 1976 



This paper is based on my Ph.D. thesis, "Market Behavior Under 
Uncertainty", M.I.T., September, 1975. I wish to thank Professors Franklin M. 
Fisher, Peter A. Diamond, Robert E. Hall, and Paul L. Joskow for valuable 
comments and advice. 



Wfi:?/ 



/VO 



, r/4 




1. Introduction 

Vertical Integration has never been a well understood phenomenon. 
Although it is possible to postulate conditions such as technological 
coordination where the incentives for vertical integration are obvious, 
it is still difficult to rationalize the amount of vertical integration 
that occurs in our economy. The dissatisfaction with the treatment 
of vertical integration is best expressed by Williamson, 

"The study of vertical integration has presented difficulties 
at both the theoretical and policy levels of analysis. That vertical 
integration has never enjoyed a secure place in value theory is 
attributable to the fact that, under conventional assumptions, 
it is an anomaly." 

It has always been somewhat of a mystery why businessmen, as well 
as researchers, so often conclude that the signficant force explaining the 
vertical integration movement has been the desire to obtain a more certain 
supply of inputs - even though these inputs appear to be sold on what most 
would call a competitive market. Why are markets not doing their jobs 
of allocating resources and why should uncertainty create incentives for 

vertical integration? 

2 
Elsewhere, I have argued that most markets do not precisely fit the 

classical requirements that prices can fluctuate instantly to equate supply 



0. Williamson - "The Vertical Integration of Production: Market Failure 
Considerations", American Economic Review , May, 1971. 

2 
D. Carlton - "Market Behavior Under Uncertainty", M.I.T. Ph.D. thesis, 

September, 1975. 



0727637 



and demand or that buyers and sellers can always buy and sell all they want 
of the good. Although ignoring these complications may be allowable for 
understanding certain types of economic behavior, the thesis of this paper 
is that it is only when the question of vertical integration is examined in 
the context of these more general and realistic competitive conditions 
(which include the classical conditions as a special case) that the in- 
centives for and consequences of vertical integration can be fully under- 
stood. Moreover, the analysis provides a justification for the frequently 
voiced claim of businessmen that they vertically integrated to obtain a 
more certain source of input supplies. 

In the next section, we discuss the previous thinking about vertical 
integration, especially regarding the effects of uncertainty. Then, 
the theory of a single competitive market under uncertainty is presented. 
Within this market framework, the effects of the transmission of uncer- 
tainty between different markets is then examined. Focusing on this trans- 
mission of uncertainty, the analysis shows that vertical integration can be 
regarded as a means of transferring risk from one sector of the economy to 
another. From a private point of view vertical integration is very attrac- 
tive. Firms have an incentive to integrate to insure a supply of input 
to satisfy their "high probability" demand. However, in a market character- 
ized by uncertainty, externalities abound, and it turns out that for the 
models under discussion, vertical integration is socially undesirable. The 
free market cannot be relied upon to achieve the socially desirable alloca- 
tion of risk and production. 

On the other hand, vertical integration does have some desirable 
attributes. We show that a vertically integrated firm is more likely to 



introduce socially beneficial technology than is a nonintegrated firm. We 
are led to a Shumpeterian view of the world where static inefficiency in 
market structure must be tolerated to obtain a type of dynamic efficiency 
in the introduction of technology. 

It is important to keep in mind throughout this paper that under 
classical assumptions about competition with constant returns to scale, 
there are no private incentives or disincentives for vertical integration, 
no undesirable social consequences for vertical integration, and no effects 
resulting from vertical integration on innovative activity. 



2. Relation to Previous Research 

The literature in industrial organization is replete with statements 
to the effect that it is the uncertainty of factor supplies that creates 
incentives for vertical integration. For example. Chandler, in his dis- 
cussion of the reasons for the formation of the largest companies in the 
United States argues "the initial motives for expansion or combination 
and vertical integration had not been specifically to lower unit costs 
or to assure a larger output per worker by efficient administration 
of the enlarged resources of the enterprise. The strategy of expansion 
had come... from the desire... to have a more certain supply of stocks, 
raw materials and other supplies..." In studying Dupont's reasons for 

integrating General Motors, Chandler finds that "the need for assured 

2 
supplies demanded increasing vertical integration. Regarding General 

Motors, we find that "Durant personally organized a number of them [i.e., 

vertically integrated] in order to make certain that his assembly line 

3 
would have a dependable supply of parts." Despite the frequency with 

which the argument about the need for assured supplies appears in historical 

studies of vertical integration, it is usually never explained why the 

factor supply is uncertain, or why an uncertain factor supply should create 

incentives for vertical integration. 



A. Chandler, Jr. - Strategy and Structure; Chapters in the History of 
American Industrial Experience , M.I.T. Press, 1964, pp. 37. 

^ibid. p. 84 

^ ibid. p. 116 



At a more theoretical level, several authors concerned with industrial 
organization have suggested that uncertainty could provide an incentive 
for vertical integration. As early as 1937, Coase argued that with 
constant returns to scale in production the very existence of a firm de- 
pends on some sort of market imperfections. The firm organizes when its 
internal allocative ability is superior to that of a market. Coase 
claimed that uncertainty about finding sellers of factors of production 
could provide one justification for the existence of a firm. Applying 

Coase' s reasoning to the question of vertical integration many years later, 

2 
Malmgrem indicated that the presence of uncertainty could create incentives 

for vertical integration. "Activities which tended to fluctuate, causing 

fluctuations in prices and outputs in the market, could be integrated 

3 
and balanced against one another." Malmgrem argued that when prices 

do not reflect scarcity, vertical integration can occur. More recently, 

4 
Williamson discussed how uncertainty can make it difficult to establish 

contracts, and could provide incentives for vertical integration. 

None of the discussions ever address the issue of whether vertical integration 

is a socially desirable response to the uncertainty in the market. 

Until very recently, there had been no attempt to analytically 

investigate the claims of the above authors as regards the effect of 

uncertainty on the incentives for vertical integration. Recently, two 

economic theorists have sought to bridge the gap in the literature. 



R. Coase - "The Nature of the Firm", Economica , November, 1937. 

2 
H. Malmgrem - "Information, Expectations, and the Theory of the Firm", 

Quarterly Journal of Economics , November, 1971. 

^ibid. 

4 
Williamson, op. cit. 



Arrow has shown that as in the deterministic case, the presence of uncer- 
tainty in a factor market with a freely fluctuating price does not create 
any incentives for vertical integration. Arrow then proceeds to investigate 
the case where informational advantages accrue to vertically integrated 
firms. In a paper closely related in topic to this one. Green showed 
that if rationing is possible in the factor market, then all firms would 
have an incentive to fully integrate. Green then goes on to examine the 
case where a vertically integrated firm is assumed to have a less efficient 
technology for producing the input. Prices are exogenous in Green's model 
so that prices need bear no relation to scarcity or rationing probability. 
In Green's model, final product firms face no uncertainty in their demand 
for their product and are able to sell all of their product at the exogenous 
market price. 

The focus of this paper is to show that in markets characterized 
by the type of uncertainty and price inflexibility discussed earlier, 
the transmission of uncertainty from the product market to the factor 
market can create strong private incentives for vertical integration 
to occur, even when such vertical integration is socially undesirable. 
In order to concentrate on the effects of the transmission of uncertainty, 
we will avoid making any of the usual assumptions that can lead to vertical 
integration. In both the final product and factor market, firms will 
compete with each other, will be untaxed, and will have constant returns to 



TC. Arrow - "Vertical Integration and Communication", Institute for 
Mathematical Studies in the Social Sciences, November, 1974. A version 
of this paper appears in The Bell Journal of Economics , Spring, 1975. 

2 
J. Green - "Vertical Integration and Assurance of Markets", Discussion 

Paper 383, Harvard Institute of Economic Research, October, 1974. 



scale in production. Integrated and nonintegrated firms will have the 
identical production technologies available to them so that there is no 
asymmetry in the production efficiency of the factor input. The prices in 
both the final and factor market will be endogeneously determined in ac- 
cordance with the analysis in the following section. Firms will have an 
incentive to integrate to lower the probability of being unable to obtain 
the factor input (i.e., to better "assure" themselves of supplies). Firms 
will have an incentive not to integrate to avoid the probability of being 
left with unused input. We see then that vertical integration involves a 
balancing of risks. Does the firm wish to bear the risk of having unsold 
inputs itself or does it prefer to let some other firm in the factor market 
bear the risk of having an unsold product? 



3. Market Behavior Under Uncertainty 

Before developing a model of how markets interact under uncertainty, 
it is first necessary to develop a theory of how a single market under 
uncertainty operates. In this section, the theory of competitive equi- 
librium under uncertainty is briefly discussed. For a fuller development 
of the theory see Carlton, op. cit. 

For most markets, prices do not adjust at each instant of the day 
to keep supply and demand in balance, firms never feel they can sell 
all they want at the going price, and production cannot occur instantly. 
More realistic assumptions would be that in order to be an effective 
"signal" prices stay fixed for some period of time. An individual firm 
never knows exactly what demand for its product will be each day, even 
if total demand for the industry is unchanging. The firm usually will 
know the probability distribution of demand it faces. Since production 
is not instantaneous, firms must make production decisions before observing 
demand, and hence take a risk of either overproducing or underproducing 
(or, more generally, of having unused or insufficient production capacity) . 
In this model, firms make decisions on price and production before their 
demand can be observed. 

Buyers know the price a firm charges, but do not know whether the 
firm has any goods left to sell at any particular time. When he goes 
to the firm, a buyer knows the probability of being able to purchase 
the good at the stated price of that firm (i.e., firms acquire reputations 
for reliability) . A good produced by a particular firm now has two 
relevant characteristics, its price and the probability that it is available. 



For simplicity, we assume no searching on the part of buyers and 
no inventory holdings on the part of firms. Provided that search cost 
and inventory holding are costly, the basic ideas of how markets operate 
under uncertainty would not be altered by their inclusion (although the 
analysis would become mathematically intractable) . We do not allow 
firms to trade the good with each other or allow recontracting markets 
to develop. This last requirement corresponds to the observation that 
in the real world, we do not always see such phenomena occuring, presumably 
because of transaction costs. 

The amount of the good that a firm begins the market period with (or 
more generally the amount of productive capacity available) will affect the 
probability that a buyer will be able to purchase the good from that firm. 
Notice that the stochastic structure of demand imposes real costs on the 
firm. Given the stochastic structure of demand, the expected profits of a 
firm will depend on the price it charges and on how many customers it is 
able to satisfy. Equivalently, we can say that expected profit depends on 
price and the probability of satisfaction. 

Letting 1-X = probability of a buyer obtaining the good and p = 
the price, it is possible to draw isoprofit curves in (1-X,p) space. 
These curves slope upward reflecting the fact that as a firm increases 
the probability of satisfying a customer by stocking more or having 
additional capacity, the price must rise to cover the Increased risk of 
having unsold goods, or available, but unused, capacity. 

On the other side o£ the market, consumers have preferences between 
the price they pay for a good and the probability that they obtain the 
good. Consumer preferences can be represented by isoutillty curves 



10 



in (1-A,p) space, where once again 1-X stands for the probability of ob- 
taining the good, and p is the price. Isoutility curves slope upward. 
We impose no restrictions on the shape of these isoutility curves other 
than the weak restriction that the slope of these curves is positive and 
bounded away from zero and infinity. 

For purposes of this paper, it suffices to consider the case where 
all firms and all customers are identical. In such markets, firms compete 
with each other on the utility level (i.e., the (1-A,p) mix) that they 
offer to consumers. Consumers will choose to frequent only those firms 
who offer the highest utility levels. Firms bid up the utility level until 
expected profits are driven to zero. Just as in other competitive markets, 
in this market equilibrium, no firm offering less than the best deal 
(i.e., best utility level) in the market will receive any customers. 
Equilibrium for such markets is depicted below as a tangency between the 
zero profit curve (tt = 0) and an isoutility curve (u(l-A,p) = u) . (With 
no additional assumptions on the shape of the isoutility curves, multiple 
tangencies and tangencies involving isoutility curves with less than the 
maximum attainable utility level are possible. For detailed discussions 
of market dynamics, nonexistence of equilibrium, and multiple equilibria, 
see Carlton, Ch. 2, op. cit.) 



Market Equlli bri urn 



Probability 

of 
satisfaction 

1 - X 



isoutility Curve 




IT = curve 



price, p 



11 



We assume for simplicity a constant cost, c, for producing one unit of 
the good. Notice that if instantaneous production were possible, there 
would be no risk that a firm need incur. In such a case, the zero profit 
curve becomes vertical at p = c, utility level competition becomes equiv- 
alent to price competition, and the equilibrium is identical to the tra- 
ditional supply and demand equilibrium. 

The noteworthy features of competitive equilibrium under uncertainty 
are that, in general, 1) the probability that a customer will be unable 
to purchase the good definitely exceeds zero, 2) the price will always 
exceed the constant marginal cost, c, of production, and 3) the total 
amount supplied and total amount demanded will not in general be equal. 

In equilibriiim, price must exceed the marginal cost of production 
since the revenue from sold goods must compensate not only for the cost of 
production of these goods, but also for the cost of production of the 
unsold goods (or equivalently , compensate for unused but available pro- 
ductive capacity) . 

The social welfare implications of markets under uncertainty are much 
different from those of markets under certainty. For example, it is pos- 
sible to show that in a two good world (where the alternative good is 
always available) , the competitive pressures will not lead to a Pareto 
efficient outcome in the sense that it is possible to organize production 
in an uncertain world so as to make consumers better off. Under plausible 



12 



conditions on preferences, it can be shown that production of the good 
subject to shortages should be subsidized. 

It is also possible to prove that as the ratio between customers and 
firms increases, the competitive equilibrium under uncertainty approaches 
percentagewise the equilibrium under certainty. In other words, as the 
customer per firm ratio increases, the equilibrium price approaches the 
constant cost of production, c, and the equilibrium probability of satis- 
faction approaches one. The ratio between supply and demand approaches 
one, while the absolute discrepancy between the two grows unboundedly. 
The rate of convergence of the competitive equilibrium under uncertainty 
to that under certainty is slow, and in general customer per firm ratios 
in excess of 5000-10000 are required before convergence to 1% of the 
deterministic equilibrium can occur. (See Carlton, op. cit. , for more 
details.) 

With this description of how single markets operate under uncertainty, 
we can now turn to the study of vertical integration as the transmission 
of uncertainty between different markets. 



we have not allowed insurance markets to develop. An insurance contract 
would provide compensation if the good were unavailable. There are 
well-known reasons why such insurance markets do not exist in the real 
world. For example, ascertaining that the customer actually made an 
attempt to purchase the good could require hug^ monitoring costs. If 
insurance markets could be costlessly run, then the private market 
would lead to an efficient outcome in this two good world. 



13 



4. The Model 

This section presents a simple model of the transmission of uncertainty 
between a product market and one of its factor markets. The model is 
intended to elucidate the incentives and consequences of (backward) 
vertical integration. 

There are two types of firms, stage 1 and stage 2 type firms. 
Stage 1 firms require factor inputs from stage 2 firms to produce the final 
good. There are N^ identical stage 1 firms and N„ identical stage 2 
firms, with N„ less than N^ . Demand facing an individual stage 1 firm 
is random during any market period. Therefore, the derived demand of 
stage 1 firms for factor inputs is random. In the stage 2 factor market, 
the demand facing any firm is also random. The final good cannot be pro- 
duced without the factor input, and the amount of the factor input avail- 
able in any period must be determined before any of the demands for the 
final product can be observed. Therefore, there is a risk that a unit of 
input will be produced but not used by the time the market period ends. We 
assume that unused input is discarded at the end of the market period. 
However, even if inventory can be held from one period to the next, as long 
as there are costs to holding inventories, the same types of qualitative 
results as developed below will hold. Prices are set at the beginning of 
each market period before any demands are observed, and are not allowed to 
vary within any market period. 

We allow stage 1 firms the option of producing some of 
the factor input for Itself. We refer to the production and holding of 
the input by stage 1 firms as vertical integration. If a stage 1 firm 



14 



produces the factor input for Itself, it bears the risk of having unused 
input at the end of the market period. A stage 1 firm is not allowed 
to sell its inputs in the stage 2 factor market. This last assumption 
is designed to capture the notion that a vertically integrated firm is 
trying to produce for its own needs to better assure itself of the supply 
of the input. Stage 1 firms cannot ship the factor input between them- 
selves, nor can stage 2 firms for the same reasons as discussed in Section 3. 
As before, we assume recontracting markets do not develop. 

We assume that the production technologies for producing the final 
product and factor input both involve constant returns to scale. The 
same technology for producing the factor input is available to both stage 1 
and stage 2 firms. It costs c to produce one unit of the factor input. 
The final product is produced (instantaneously) by a Leontief technology 
that requires K units of capital and 1 unit of the factor input sold in the 
stage 2 market to produce one unit of the final good. The capital input is 
always available at a constant price r per unit. 

The market operates in the same manner as described earlier. There 

are assumed to be L identical customers. In each market period, each of 

2 
the L customers randomly frequents one stage 1 firm where he demands the 

final product according to his demand curve. Every time a stage 1 firm 

observes a customer demand for its product, it attempts to obtain the 

factor inputs necessary to produce the customer's demand for the final 

product. The stage 1 firm first tries to use up its own holdings, if any. 



See next page for footnote. 



noTe precisely, each customer randomly frequents one stage 1 firm from 
among those firms which he feels are offering the best deals in the 
market . 



14a 



Let me give two possible reasons and examples to illustrate why this 
assumption is reasonable. First, an output firm might be distrustful 
of the quality of an input purchased from another competing output firm. 
Second, there may be transaction costs or other increases in bureaucratic 
costs as a company expands from being a seller of output to becoming 
a seller of both outputs and inputs. Complex organizations can increase 
costs. 

This nonsharing of an input supply by output producers is quite 
common. As an example consider a firm's secretarial pool that is occa- 
sionally idle. It is not frequently the case that an employer will try 
to obtain outside typing in order to keep his secretaries fully busy. 
Next, consider a firm that keeps stocks of some input for production. 
This firm will usually not be always willing to sell its input. The 
firm will usually enter the input market as a seller only if its stock 
of input starts piling up and the firm becomes convinced that it miscal- 
culated its needs. Only when the firm becomes sure that it has made a 
sizable miscalculation will it pay for the firm to incur transaction 
costs and enter the input market as a seller. 

A model with equivalent implications can be constructed by assuming 
that vertically integrated firms can purchase input from each other, 
but the input Is also demanded by other sectors of the economy, and that 
vertically integrated firms do not sell to these other sectors. 



15 



of the factor input, and then, when its factor holdings are depleted, it 
enters the stage 2 factor market. Once in the factor market, the stage 1 
firm randomly frequents a stage 2 firm to try to obtain the necessary 
inputs to be able to satisfy its customer. If the stage 1 firm is unable 
to obtain the input from the stage 2 firm, then the stage 1 firm is unable 
to satisfy the demand of the customer. This customer returns home dis- 
satisfied. As discussed in Section 3, customers have preferences, which 
firms recognize, between the price of the good and the probability of 
obtaining that good. For any given level of factor holding by the stage 1 
firms, we can imagine the stage 1 and stage 2 firms competing in their 
respective markets on the price and probability of satisfaction until each 
market reaches the competitive equilibrium described in Section 3. 

The important feature of this market structure is that the amount of 
the factor input that stage 1 firms decide to hold affects the stochastic 
nature of the demand that stage 2 firms see. Vertical integration by 
stage 1 firms affects the risky environment in which stage 2 firms operate. 



16 



5. Issues Associated with Firm Interaction Under Uncertainty 

Now that the model of the transmission of uncertainty between the 
stage 1 and stage 2 markets has been described, we can state more clearly 
the issues that we wish to examine. The decision of stage 2 firms about 
how much of the input to produce affects the probability that a stage 1 
firm will be able to obtain the input and produce the final product. 
If a stage 1 firm becomes dissatisfied with the operating policy of stage 2 
firms, the stage 1 firm can produce some of the input for itself, and itself 
bears the risk of having unsold input at the end of the market period. 
The decision of stage 1 firms to produce some of the input for themselves 
and bear the risk of having unsold input, affects the entire stochastic 
structure of demand that the stage 2 firms will see, and hence influences 
the equilibrium that is reached in the stage 2 market which, in turn, 
influences the equilibrium that is reached in the stage 1 market. 

Firms in each market compete amongst themselves until equilibrium 
is reached in the manner described in Section 3. Therefore, we know that 
in the equilibrium in both the stage 1 and stage 2 markets, each firm's 
operating policy reflects the preferences of its customers, and that the 
prices reflect the probability of obtaining the good. The important 
question to ask is whether, under competition, firms are forced to take 
into full account the effect of their operating policies on the transmission 
of uncertainty to other markets. What happens to the welfare of consumers 
as final product firms produce some of the factor input for themselves 
and themselves bear the risk of having unsold input? From the consumers' 



Recall from Section 3, that the stochastic structure of demand affects 
the operating cost of firms. 



17 



point of view, is there some preferred allocation between the stage 1 
and stage 2 firms for producing the input and bearing the risk of having 
unsold inputs? Do the incentives under competition lead firms to adopt 
this preferred market structure, or, are the incentives under competition 
perverse, and discourage firms from adopting the preferred operating 
policies? If competition does not lead to the preferred market structure, 
how does the competitive equilibrium differ from the preferred one? 
If suddenly a new technique for producing the output becomes available, is 
it more likely to be adopted when the market structure is integrated or 
nonintegrated? 



18 



6. Private Incentives for Vertical Integration 

The way the market operates, there are basically two offsetting 
considerations Involved in the decision of a stage 1 firm to produce 
a unit of input for Itself. 

First, since it costs only c per unit to produce the input, the stage 1 

firm will save (p. - c) by producing the Input itself rather than buying 

it on the factor market at price p. . (Recall that it is a characteristic 

int 

of markets that operate under uncertainty that the equilibrium price 
exceed the cost of production.) In other words, if the stage 1 firm 
produces the input itself, then the firm assures itself of having the 
necessary input to make a more profitable sale, if demand should materialize. 
Offsetting this saving is the potential risk that the input will be pro- 
duced at cost c, but will not be used because of insufficient demand. 
By producing the input for itself, the stage 1 firm bears the risk of 
the unsold input, while when the stage 1 firm relies on the factor market 
for the input, it is the stage 2 firms who bear this risk. 

Let us now consider whether there is any incentive for stage 1 firms 
to produce the input at all. Suppose that stage 1 firms produce none of 
the factor input and rely entirely on the stage 2 markets for the input. 
Imagine that the stage 1 and stage 2 markets have reached equilibrium in 
the manner described earlier. By the conditions of market equilibrium, 
expected profits are zero in each market. 

There will definitely be an incentive to vertically integrate if the 
expected profit is positive when a stage 1 firm holds enough Input to 
satisfy one customer when evaluated at the equilibrium associated with 



19 



no vertical integration. It is straightforward to calculate the profit 
functions, and to derive the condition that there will definitely be an 
incentive to vertically integrate if 



[1 - Pr(0)] p. > c , where 
int 



[1 - Pr(0)] = probability that at least one customer frequents 
the stage 1 firm, 

p . = price of the intermediate product when purchased 
in the stage 2 market, and 

c = cost of producing the input. 



This inequality is intuitively plausible. If a stage 1 firm decides 
to stock one unit of input, its cost increases, with certainty, by c. 
Its expected savings from not having to go into the factor market for 
that one unit of input is [1 - Pr(0)] p. . When savings exceed costs, 
the stage 1 firm will hold the factor input. 

It is possible to simplify this inequality. Since the random process 

is binomial with probability — -, and size L, we have that [1 - Pr(0)] = 

T L -L/N, 1 
1-(1-— ) =l-e . Therefore, we can write that there will 

definitely be an incentive for vertical integration if 

-L/N^ 
(1 - e ■") p^^^ > c . (1) 

2 
It is clear from the inequality (1), that for fixed L, the inequality 

is more likely to hold the smaller is the number, N^ , of stage 1 firms. 



See Appendix 1. 



2 
It is not possible to determine whether (1) is more likely to hold as the 

-L/Nj^ 

number of customers increases to infinity. As !->■<», 1-e ->-lso 

incentives to vertically integrate increase. However, as L ->■'», p. j. ■*" c, 

so disincentives also increase. Which effect will predominate depends on the 

specific slope of the isoutility curves. 



20 



Equivalently, for a fixed number of customers, L, the inequality is more 
likely to hold as the customer per store ratio L/N, becomes larger. 

This last result emphasizes the importance of examining the uncertainty 
in the market clearing process. Earlier, it was mentioned that for large 
customer per firm ratios, the market equilibrium under uncertainty ap- 
proached that under certainty at least percentagewise. In such a case, one 
might have thought that the deterministic analysis which ignored the uncer- 
tainty in the market would suffice, and the more complicated analysis which 
explicitly considered the uncertainty was unnecessary. What we see here 
though is that it is precisely the case of having a large customer per 
firm ratio in stage 1 markets that can lead to strong incentives for 
vertical integration. A deterministic analysis of this market structure 
would have been unable to find any incentives or disincentives for vertical 
integration to occur. 

The implications of (1) are disturbing. For any value of L/N, , we know 
that incentives for vertical integration can exist since it is always 
possible to choose a set of preferences which yield an equilibrium P. ^ 
such that the inequality holds. Moreover, for even small values (e.g., 
20) for the customer per firm ratio, L/N, , 1 - e 'fe 1, so that the in- 
equality will hold provided p. > c. But, by the earlier discussion, 
we know that the equilibrium price in an uncertain market must always 
exceed its cost, since it is necessary that price not only cover per 
unit production costs, but also the production cost of unsold goods. 
Hence, we expect there to be strong incentives for vertical integration. 

The incentives for vertical integration come about because the 
stage 1 firms base their decisions to integrate on the marginal , not 



21 



average , probability of using an additional input. The way the markets 
operate, the price of the factor in the stage 2 market reflects not only 
the cost c of producing the input, but also the average probability of 
not being able to sell that input. When a stage 1 firm is deciding whether 
to hold one unit of the input itself, it is not concerned with the average 
probability of being unable to use any unit of input. Rather, since the 
stage 1 firm will use its input holdings first, the stage 1 firm is con- 
cerned with the probability of being able to use that first unit of input. 
For even low to moderate values of the customer per firm ratio in stage 1 
markets, L/N, (e.g., 15-20), this probability is practically 1 (i.e., 
each stage 1 firm is virtually assured of being able to use up its one 
unit of input) , so that (1) will almost certainly hold since the price 
of the stage 2 factor, p. , exceeds c. It is precisely because stage 1 
firms can use their own input to satisfy their "high probability" demand 
and use the stage 2 market to satisfy their "low probability" demand 
that incentives for vertical integration occur. The conclusion of this 
analysis is that it is quite likely that there will exist strong private 
incentives for vertical integration to occur. 



22 



7. Social Consequences of Vertical Integration 

Having established strong private incentives for at least some 
vertical integration to occur, we now investigate the welfare consequences 
of vertical integration. It is useful to keep in mind that there are 
two distinct welfare issues involved. First, as mentioned earlier, 
markets under uncertainty are not Pareto-ef f icient in the absence of in- 
surance markets, and usually lump sum subsidies are required to achieve 
optimality. The second issue is whether a vertically integrated market 
structure is superior to a nonvertically integrated one, whether or not 
lump sum subsidies are paid. It is this second issue that we now address. 

If all firms within any stage behave identically, then there is always 
a higher probability that a unit of the factor input will be used if 
it is held in a stage 2 rather than a stage 1 firm. Stated in another 
way, since the number of stage 1 firms exceeds the number of stage 2 
firms, a unit of the factor will be more frequently used if it is given 
to a stage 2, and not a stage 1, firm. From this simple observation, we 
can obtain the following. 

Theorem 1 ; Any market structure involving vertical integration achieves 
a lower level of expected utility than can a market structure involving no 
vertical integration. 

The reasons why Theorem 1 is true can be explained intuitively as 
follows. The number of final product stage 1 firms exceeds that of factor 
input stage 2 firms. Therefore, stage 2 firms are more efficient absorbers 



An analytic proof appears in Carlton, op. cit. 



23 



of risk in the sense that stage 1 firms would have to hold more of the 
input than stage 2 firms in order to satisfy the same fraction of the popula- 
tion. Although holdings of the input by stage 1 firms reduces the demand 
seen by stage 2 firms, this reduction in demand is not great enough to 
offset the inefficient risk absorption by stage 1 firms. Therefore, stage 2 
firms must decrease their input holdings by less than the amount that stage 1 
firms increase their input holdings, if the same fraction of the population 
is to be satisfied. So, when stage 1 firms produce any input for themselves, 
more total input in the system must be produced or the fraction of customers 
who are satisfied will decline. To satisfy any given fraction of consumers, 
market structures involving vertical integration will have higher input 
costs than those involving no vertical integration. Since competition 
insures that cost savings are passed on to consumers, it follows that 
consumers can always be made better off whenever there is any vertical 
integration in the system. 

Theorem 1 can be heuristically explained in terms of sharing. 
Consider the following example. There are two bakeries, side by side, 
and 100 customers who each day randomly frequent one bakery and buy one 
loaf of bread. If the bakeries are willing to share their production of 
bread with each other, then only 100 loaves need be produced to satisfy 
the entire population. If the bakeries refuse to share (or else if it 
is costly to share) with each other, then each bakery must produce 100 
loaves of bread to be sure to satisfy the customer population. Sharing 
allows 100, instead of 200, loaves to suffice. Exactly analogous reasoning 
applies to vertical integration. When stage 1 firms produce the input 
for themselves, they in effect do not share it with other stage 1 firms. 



24 



However, when all stage 1 firms rely on stage 2 firms for the factor 
input, they are essentially "sharing" from common resource pools. Since 
insurance-like costs can always be lowered when no sharing is taking place, 
vertical integration can impose unnecessary costs on society. 



25 



8. Equilibrium Market Structure 

Let us now turn to a discussion of the equilibrium market structure. 
We say that a market structure is in equilibrium if at the current level 
of vertical integration, stage 1 firms have no incentive to alter the 
amount of input that they produce for themselves, and if the stage 1 
and stage 2 markets are equilibrating in accordance with the principles 
discussed earlier of how markets clear under uncertainty. There are 
basically 3 types of possible equilibrium market structures. First is 
the case of no vertical integration. The second and third types of market 
structures both involve vertical integration. We call the second type 
of vertical integration, partial vertical integration. As its name suggests, 
in this market structure, stage 1 firms are vertically integrated, but 
still rely on stage 2 markets to provide some inputs. The third type of 
market structure is that of complete vertical integration, in which each 
stage 1 firm relies only on itself for its supply of the input and the 
stage 2 market disappears completely. 

If the equilibrium market structure is of the type involving incom- 
plete vertical integration, then the stage 2 market acts like an insurance 
market for supplying the factor input to the stage 1 market. To see this 
last point, notice that whenever a stage 1 firm makes a sale of a final 
product, it makes a higher per unit profit when it is able to use its own 
input (produced at cost c) in the manufacture of the final good rather than 
when it uses an input purchased on the stage 2 market at a price p. 
(which exceeds c) . A stage 1 firm continues to enter the stage 2 market 
simply because it needs to satisfy its customers. It is cheaper for a 



26 



stage 1 firm to satisfy its customer through use of the high price stage 2 
market, rather than produce extra input for itself and bear the risk that 
the unit of input will go unsold. 



27 



9. The Consequences of Vertical Integration on Market Equilibrium 

In this section, we examine how the market equilibrium is affected 
when the equilibrium market structure involves some vertical integration. 
How does the market equilibrium with vertical integration compare to the 
market equilibrium when vertical integration is not allowed? From Theoren 1, 
we already know that, compared to the expected level of utility achievable 
in a competitive equilibrium with no vertical integration, the expected 
level of utility is lower in any market structure involving vertical in- 
tegration. Since vertical integration lowers the maximum expected level of 
utility, it is possible that the level of utility could be driven so low 
that consumers would prefer not to enter the stage 1 market. Thus, one 
consequence of vertical integration can be to drive output markets out of 
existence. Having mentioned this possibility, we shall concentrate in the 
subsequent analysis on the effects of vertical integration when the markets 
under study remain in existence. 

The questions we ask are whether the price, p , , of the final product, 
the price, p. , of the factor sold in the stage 2 market, and the probability 
of satisfaction, 1 - A, are higher or lower in a vertically integrated 
market structure than in a market structure in which vertical integration 
is not allowed. This section states the main results and provides explana- 
tions for them. The reader interested in the technical proofs is referred 
to Carlton, op. cit. 

Suppose that the equilibrium market structure involves complete 
vertical integration. Under a plausible assumption on preferences. 



footnote on next page 



27a 



The assumption is analogous to the assumptions of "normal" goods in 
economic theory. We assume that if the customer per firm ratio increases 
so that the consumers are offered a better "menu" of price-probability 
of satisfaction combinations, then the consumer will prefer a combination 
in which he is made better off in both dimensions. (As the customer 
per store ratio increases, the zero profit curve, used to define equilibrium, 
shifts up so that consumers are faced with a better set of price-probability 
of satisfaction choices.) In other words, the consumer will choose 
a combination with a lower price and higher probability of satisfaction. 
This assumption appears very reasonable since it is possible to prove 
(see Carlton, op. cit. ) that as the customer per firm ratio increases, the 
market equilibrium price approaches its minimum possible value of c, and 
the probability of satisfaction approaches its maximum value of one. 
Hence the consumer must be made better off in terms of both the price and 
probability of satisfaction for sufficiently large increases in the customer 
per firm ratio. To see the analogy of the above assumption to consumer 
theory, note that we usually assume that a consumer in a two good world 
would increase his consumption of both goods in response to an increase in 
income (i.e., as his "menu" between the two goods improves). We also 
make an assumption that the isoutility curves are of such a shape that 
multiple tangencies (i.e., multiple equilibrium) with the zero profit 
curve do not occur. 



28 



it is possible to establish that the market equilibrium with complete 
vertical integration involves a higher price and lower probability of 
satisfaction than does the market equilibrium with no vertical integration. 
Stage 1 firms are less efficient absorbers of risk than stage 2 firms 
in the sense that stage 1 firms with complete vertical integration have 
to spend more resources on production of inputs than do stage 2 firms, 
with no vertical integration, to satisfy any given fraction of the popula- 
tion. The final price to the consumer has to rise to cover this increased 
cost of operation in the case of complete vertical integration. Moreover, 
because stage 1 firms cannot satisfy customers as efficiently as stage 2 
firms, the equilibrium probability of shortage in the case of complete 
vertical integration rises from its value in the market equilibrium when 
vertical integration is not allowed. From these two results, it also 
follows that the total amount of the output that is purchased is lower 
in the case involving complete vertical integration than in the case 
involving no vertical integration, provided per capita demand depends 
negatively on price. 

Suppose now that the equilibrium market structure involves partial 
vertical integration. The main result is that the price of the input 
purchased in the stage 2 market is higher in the case involving partial 
vertical integration than in the market equilibrium when no vertical 
integration is allowed. This result follows from the fact that in the 
case of partial vertical integration, the stage 2 markets become "riskier" 



"Risk absorbing" efficiency is inversely related to the customer per firm 
ratio. When stage 1 firms produce some of their own input, the stage 2 
firms see less than L customers, so that the customer per firm ratio of 
stage 2 firms falls from its value in the case of no vertical integration. 



29 



and the stage 2 firms become less efficient absorbers of risk than they 
were in the case of no vertical integration. This inefficiency results 
in increased costs to stage 2 firms. To cover their increased costs, 
the stage 2 firms have to raise their prices to the stage 1 firms. Sur- 
prisingly, it does not appear possible to prove that the stage 1 firms 
pass this increased cost along to the consumer in terms of higher prices 
for the final good. It seems possible, though I suspect unlikely, that 
with partial integration, the price of the stage 2 input could rise, but 
the price of the final good could fall. In this case, we know from Theorem 1 
that the probability of satisfaction would have to fall sufficiently so 
that consumers are worse off in the case of partial vertical integration 
than in the case of no vertical integration. 

In summary then, any market structure involving vertical integration 
provides a lower level of utility to consumers than can a market structure 
involving no vertical integration. Any vertical integration causes an 
inefficiency in the ability of firms to absorb risk, and usually will 
result in higher prices in the input market. When the equilibrium market 
structure involves complete vertical integration, we expect that both 
the probability of shortage and the price of the final good will rise 
from their equilibrium values in the case when integration is not allowed. 



30 



10. Market Structure and the Choice of the Output Technology 

In this section we examine how market structure and the transmission 

of uncertainty between firms can influence the choice of technology. 

So far, we have assumed that there is only one technology to produce the 

output, namely a Leontief technology which uses K units of capital and 

one unit of the input subject to shortages to produce one unit of output. 

Now we will assume that there suddenly becomes available a new Leontief 

technology with input requirements (K^,£). We examine the incentives 

for introduction of the new technology in a nonintegrated and integrated 

market setting. The main conclusion of this section is that introduction 

of a new technology that would benefit society is more likely to occur 

in a market with vertical integration than in one without vertical integration. 

In the discussion of market clearing, it is useful to introduce 

the concept of a "derived" isoutility curve. A derived isoutility curve 

reflects trade-offs between the price of the input and the probability 

of obtaining the input that translate (through the stage 1 zero profit 

condition) into trade-offs that the consumer is willing to make between 

the price of the final good and the probability of obtaining the final 

good. A simple example is the best way to illustrate this concept. 

Consider the case of no vertical integration. Assuming Leontief 

technology with coefficients (K,l), the zero profit condition for stage 

1 firms is simply p . = rK + p . , where p^ is the price of the final 

f int f 

good, p. is the price of the input sold on the stage 2 market, and 
r is the price of capital. Preferences of consumers are represented 
by u(l - A,p^) = u. Substituting in for p- from the zero profit condtion. 



31 



we find that u(l- X, rK + p. ) = u now expresses the consumer trade-offs 

int 

in (1 - X, p. ) space. Equilibrium in the stage 2 market is determined 
by the tangency between this derived isoutility curve and the stage 2 zero 
profit curve. 

The important feature about derived isoutility curves is that their 
shape will be influenced by the input-output coefficients of the tech- 
nology. So, for example, if we had a new technology (K ,Ji) where £ < 1, 

any increase in p. would translate into a smaller price increase in p^ 
int f 

than it would if £ = 1. In this case, the derived isoutility curves 
would become flatter than they are in the case of £ = 1. We will refer 
to the original (K,l) technology as the "original" one, and the (K^,Jl) 
technology as the "new" one. 

First, consider the market equilibrium that would occur if vertical 
integration is not allowed. Let the equilibrium factor price be p? . 
The stage 1 firms will adopt the new technology only if it is more effi- 
cient when the price of capital is r and the price of the input is p? . 
But this marginal calculation is not sufficient to guarantee that consumers 
would not be better off under the new technology. The diagram below 
illustrates this point. 



Probability 
of 

satisfaction 
1 - X 



Choice of Technology 



^^3 V 




H2 = 



•derived isoutility curves - new 

technology 

derived isoutility curve - original technology 



price of the input, p^.^^ 



32 



Point E is the original market equilibrium. The derived isoutility 
curve through point E is drawn using the input-output coefficients of 
the old technology. The level of utility achieved by consumer along 
this curve is u. The derived isoutility curve, corresponding to the 
same u when the input-output coefficients of the new technology are used, 
is drawn as a dotted line. The fact that the dotted curve passes above 
point E is equivalent to the statement that the new technology is less 
efficient than the old technology at the factor price associated with 
point E. The two derived isoutility curves cross at point B, where 
each technology is equally efficient (i.e., p. + rK = Jlp. + rK,). 
Beyond point B, the new technology is more efficient. Notice that the 
dotted isoutility curve crosses the zero profit (it„ = 0) curve. Therefore, 
there exists some point C of tangency between the derived isoutility curve 
with the new technology and the tt = curve that represents a level of 
utility above u. 

Consumers would be better off if all stage 1 firms adopted the new 
technology so that the market equilibrium would move to point C. Yet, 
because stage 1 firms have no control over the input market, they will 
not have any incentive to adopt the new technology, since it is inefficient 
at the initial market equilibrium E. The existing prices do not provide 
incentives for stage 1 firms to change technologies, nor for stage 2 
firms to alter their behavior. 

It is easy to see how vertical integration could remedy this situa- 
tion. To make the point, it suffices to consider the case of complete 
vertical integration. Since each stage 1 firm totally controls its produc- 



33 



tion of the input, it can coordinate its (p. ,1 - X) mix to its own 

mt 

specifications. Because of this possibility of coordination, stage 1 
firms will be able (and through competition will be forced) to move 
immediately to any achievable point that justifies the use of the new 
technology and makes consumers better off. 

It is precisely because vertically integrated firms can exercise 
control over the characteristics (p. ,1 - X) of the input, that they 
are able to introduce the new and more desirable technology. With no 
vertical integration, price signals are not sufficient to convey the 
benefits of switching to the new technology. The reason is that whether 
consumers would be better off if all stage 1 firms adopted a new technology 
is a nonmarginal change. In this case, marginal incentives at the firm 
level do not give the correct signals as to whether the nonmarginal change 

is desired. Equivalently , in a competitive market, marginal incentives are 

2 
not always sufficient to insure that equilibrium is at a global optimum. 

In general, when the choice of technology affects the equilibrium 

characteristics (e.g., price, probability of satisfaction) of the input 

which, in turn, influence the choice of technology, then it is not necessarily 

true that the individual decisions by firms will lead to the correct 



In the case of complete vertical integration, the stage 1 firms effectively 
act as stage 2 firms since they produce their own input. 

2 
i.e., if input firms "experiment" by moving marginally around the initial 

equilibrium, they are driven back to the Initial equilibrium. The 

reader should notice that there Is nothing in this section that uses 

the fact that there is uncertainty In the market. The conclusion Is 

the general one that vertical Integration can be one way of remedying a 

market failure that occurs when marginal and global incentives differ. 



34 



technology being adopted. In such cases, existing prices need not provide 
the correct signals for choice of a new technology. In the model of this 
paper, vertical integration is a mechanism by which individual firms 
can gain control over the two characteristics (i.e., price and probability 
of satisfaction) of the input that influences its choice of output tech- 
nology. The final product firm that is vertically integrated is thereby 
better able to coordinate its choice of output technology with the character- 
istics of the input. With the flexibility of tailoring the input character- 
istics to its needs, the vertically integrated firm may introduce the new 
more desirable technology, while the nonintegrated firms who must take 
the input characteristics as given in the marketplace may get locked into 
the old technology and have no incentive to change production technologies. 

In markets characterized by uncertainty. Theorem 1 proved that vertical 
integration can be socially undesirable. A more efficient market structure 
will usually involve no vertical integration. However, socially desirable 
technologies are more likely to be developed and introduced in a market 
structure involving vertical integration in which individual firms can 
coordinate input characteristics with their choice of technology, than in a 
market structure involving no vertical integration in which such individual 
firm coordination is impossible. We are led to the Schumpeterian view of 
the world that it may be necessary to tolerate some static inefficiency in 
market structure in order to create an environment in which socially de- 
sirable technologies can be developed and introduced. 



35 



11. Horizontal Merger and Social Welfare 

Since the number of stage 1 and stage 2 firms is exogenous to this 
form of the model, one has to be very cautious about welfare interpreta- 
tions when the number of firms change. One important caveat associated 
with the model deals with the implications of horizontal merger. In the 
model, it appears that the utility of consumers increases as the number of 
stage 2 firms declines to one. One has to be careful to avoid the infer- 
ence that for the markets under study total horizontal integration of stage 2 
firms is always desirable. The model is designed to study the transmission 
of uncertainty between firms in a competitive environment. However, large 
horizontal mergers could create monopoly power which by itself entails 
social costs. Horizontal integration may be desirable from the point of 
view of risk sharing (which is what the model is designed to focus on) , but 
undesirable from the point of view of creating monopoly power. Whether 
there are any private incentives for horizontal merger to occur is an 
altogether different question. As discussed in Carlton, op. cit. , it is 
possible to envision situations where there will be no incentives for 
horizontal mergers. Possible situations are when a) demand is spatially 
random so that merger at one place can reduce demand, b) there exist 
costs to merger, c) horizontal merger is legally prohibited. Moreover, 
as we shall soon see, the incentive for horizontal integration is not 
necessarily a natural feature of markets in which random demand creates 
incentives for vertical integration. 

This paper has shown how a nonintegrated competitive market structure 
can be socially preferred to an Integrated one. However, it is not true, 
for the model under discussion, that the nonintegrated market structure 



36 



is socially optimal. As mentioned earlier, when dealing with markets under 
uncertainty, in the absence of insurance markets the social optimum can 
involve paying lump sum transfers to the firms producing the good subject 
to shortages. Therefore, the socially optimal solution would be to have a 
nonintegrated market structure and usually to pay lump sum subsidies to the 
stage 2 firms. These subsidies would usually be used to encourage the 
stage 2 firms to expand production of the input that is subject to short- . 
ages. 



37 



12. Extensions, Interpretations, Evaluation 

It Is easy to extend the model to the situation where the stage 2 
Input can be sold to demanders who are not stage 1 firms. The incentives 
for vertical integration diminish as the stage 2 market expands (since 
Pj ,. approaches c, and 1 - X approaches 1). The incentives increase if 
the demanders of the stage 2 input have very different preferences than 
stage 1 firms. In general, we expect firms to be less likely to vertically 
integrate when they form a small part of the total demand for an input. 

It is straightforward to interpret the "input" in the model as either 
capital or labor. The decision to vertically integrate then corresponds 
to decisions about labor or capital "hoarding" to insure that demand, if 
it materializes, can be met. The assumption that vertically integrated 
firms do not sell their inputs on the stage 2 market can be thought to 
correspond to either fixed contracts with no recontracting markets, or 
to the fact that it does not seem to occur frequently that firms would 
sell their capital or laborers to someone else within a market period. 

It is useful to point out that there is no distinction in the model 

2 
between "long-term" contracts at price c and in-house production at 

cost c. Clearly vertical integration can occur by either mode. As 

Williamson (1971) argues, uncertainty about supplier reliability or 



For a treatment of labor hoarding along these lines, see R.E. Hall, 
"An Aspect of the Economic Role of Unemployment", Proceedings of the 
International Economics Association , April, 1975. 

2 
A "long-term" contract is a one period contract that states that an 

output firm will accept delivery of a certain amount of input from an 

input firm. 



38 



other types of transaction costs can explain why in the real world vertical 
integration often takes the form of in-house production, and not long- 
term contracts. Notice that this consideration of which form vertical 
integration should take is much different from the issues studied in 
this paper of whether there is an incentive for vertical integration 
(whether by long-term contract or in-house production) to occur at all. 

A belief in perfect competitive markets leads one to the conclusion 
that vertical integration does not have any desirable or undesirable 
features. A belief in competitive markets which possess the demand un- 
certainty and price inflexibility discussed earlier leads one to very dif- 
ferent conclusions, as the simple model just presented illustrates. 
Which of the two views of competitive markets is appropriate will depend 
on the particular industry under study. 

Sometimes models can be misleading because of the apparent simplicity 
or form of their assumptions. Let me now restate the four key features 
of the markets under study in an effort to distinguish the features 
that are important from those that are merely analytically convenient. 

First, firms must never feel they can either buy or sell all they want 
at the going market price. In the model, the corresponding assumption was 
that firms faced demand and supply uncertainty. Not allowing firms to 
trade among themselves or allowing recontracting markets to develop 
was one simple way to characterize this "friction" in the market. Clearly, 
the qualitative features of market operation are unchanged if we allow 
firms to trade a "little" among themselves - while if firms trade "a 
lot" among themselves (i.e., recontracting markets exist), then we get 
right back to a perfect competitive market. 



39 



The second main feature of the markets under study really follows 
from the first feature. We require that there be some risk in any period 
that resources will not be fully utilized. The impact of this statement 
is that for many markets some unemployment of resources is a natural 
consequence of market operation. 

The third feature involves the differential risk that a vertically 
integrated firm can impose between the use of its own inputs and the 
inputs of a factor market firm. Because a firm will always choose to 
use its own inputs first, there is always a higher probability that a 
firm will use a unit of its own input than a unit of input that factor 
market firms hold. 

The final feature of the markets under study is that a vertically 
integrated firm be somehow less able than a factor market to satisfy 
input demands of other firms. In the model, we capture this feature 
by the (somewhat extreme) assumption that vertically integrated firms do 
not provide inputs to other firms. Clearly, once again, the qualitative 
results would be unchanged if we let vertically integrated firms do "some" 
selling on the input market. We know that for some industries such trading 
does occur, while for others it very rarely occurs, presumably for the 
transaction cost argument given in Section 4. The real question of course 
is how much trading occurs. If there is "a lot" of trading, then once 
again, we approach a perfect input market and we come back to the classical 
view of vertical integration. On the other hand, if vertically integrated 
firms will incur the necessary transaction cost and sell their Inputs (or 
equivalently use their unused capital to produce inputs) to other ccwn- 
petitors only if the firm feels that it miscalculated and has little immediate 
hope of using that input in satisfying the demand for its final product. 



40 



then we are driven to the view of vertical integration that this paper 
presents. 

It is important to stress what features of the model are not important 
and are designed only to make the model easy to use. First, the total 
industry demand need not be fixed. All that is required for the results 
of the model is a random per firm demand, which is certainly implied 
by a random industry demand. Second, the welfare implications of the 
model derive from the assumption that vertically integrated firms cannot 
satisfy input demand as efficiently (from a risk sense) as input firms. 
The assumption that the number of stage 1 firms, N, , exceeds the number 
of stage 2 firms, N„, is a convenient way of capturing this feature in 
the simple model. (Notice that in the simple model if N, is less than 
N„ then the factor market can never exist, and the simple model becomes 
uninteresting.) This assumption about the relative magnitudes of N^ 
and N„ is not meant to be an assumption about the relative numbers of 
establishments in each stage, but rather to be an assumption about the 
relative ability of each stage to absorb risk. Clearly, an alternative 
model which can lead to identical welfare conclusions is to have the 
factor market able to satisfy input demanders from other sectors of the 
economy while vertically integrated firms could not. This alternative 
model could postulate that each output firm obtained some share of total 
random demand. The input firms then obtain some share of random derived 
demand. Moreover, there are other sectors of the economy whose demand 
for input is also random and uncorrelated with the random input demand 



To preserve the competitive environment, we simply require that the 
share of demand depends on the utility level offered. As in all com- 
petitive equilibria, if a firm does not offer the best deal (i.e., highest 
utility level) its demand goes to zero. 



41 



of the output industry under study. (To avoid complications about ex- 
ternalities, postulate that output firms in all sectors have similar 
preferences and similar random demands.) Input firms can pool risks, 
while vertically integrated firms are, by assumption, unable to. With 
this new model, we obtain identical insights as before into the incentives 

for and consequences of vertical integration. Notice that in this al- 

2 
ternative model there are no incentives for horizontal integration. 

As discussed earlier, incentives for horizontal integration are not 
a necessary consequence of market forces that produce incentives for 
vertical integration. 

The simple model used in this paper has the advantage that it is 
simple enough to use analytically (for example to derive the incentives 
for vertical integration) , and yet maintain the four key features of 
market operation mentioned above. Given these four main features of 
market operation, the strong incentives for vertical integration arise 
because of the differential risk that a vertically integrated and input 
market firm face. The undesirable welfare implications of vertical 
integration result because the vertically integrated firm is a less 
efficient satisfier of input demand than is an input firm. It is in- 
correct to interpret the model as saying that all vertical integration 
is inefficient. Rather the model shows that in a world of uncertainty 
there can exist very strong private incentives for vertical integration 
to exist, even in cases where such vertical integration may be socially 
undesirable. 



The assumption that vertically Integrated firms do not sell their input 
to other sectors of the economy is based on the same transaction cost 
type arguments as expressed in Section 4. 

2 

I thank Paul Krugman for this observation. 



42 



The question then arises as to whether the incentives and consequences 
that this paper identifies are a significant feature of market operation. 
The answer to this question will obviously depend on the particular market 
under study. However, it does seem that the model fits in well with de- 
scriptive studies of individual industries. 

In an in-depth study of the automobile industry. White examines the 
reason why auto companies vertically integrate, and, most relevant for this 
discussion, provides a descriptive explanation of how risks motivate ver- 
tical integration. White's descriptive discussion echoes many of the 
points raised earlier. 

We argued that vertical integration was a means of transferring 
risk between firms. White states ". . .integration is a two-edged sword. 
Though it reduces the risk of supply failure, it also converts variable 

costs into fixed costs - . . .More money is at stake,. . .the financial 

2 
penalties of losses (that is, risks) have increased." 

We found that there would exist strong private incentives for vertical 

integration to occur, and identified the possibility for partial vertical 

integration. The strong incentives for vertical integration arise because 

the vertically integrated firm is able to satisfy its high probability 

demand by itself, and pass on the low probability demand to some other 

firm. We found that for the case of partial integration, the factor market 

acted as a type of insurance market for the final product firm, with the 

final product firm making less of a profit on any item that used a factor 

market input than on an item that used an internally supplied input. 



L. White - The Automobile Industry Since 1945 , Harvard University Press, 
Cambridge, Mass., 1971. 

^ibid., p. 80 



43 



On these issues, White writes, "A way of reducing the risks of ver- 
tical integration is through partial or tapered integration: a company can 
produce a portion of its needs of an item and buy the fluctuating remainder. 
This has the advantage of providing full utilization of its own equipment 
and allowing the suppliers to absorb the risk of fluctuations in demand. 
The company has to pay a premium to get someone else to absorb the risks, 
the the risk transfer is achieved. In the case of a supplier failure, 

production of the final good does not have to cease. . ." "Tapered 

2 
integration plays a large role in the industry." 

Moreover, as mentioned at the outset of this paper, businessmen fre- 
quently say that they vertically integrated to obtain a more certain supply 
of inputs. Based on such statements and descriptive studies like White's, 
it does appear that the incentives identified in models of market behavior 
under uncertainty do exert a significant influence on market outcomes for 
certain industries. 

The results of this research emphasize the importance of distinguishing 
between market clearing under certainty and under uncertainty. An analyst 
using a deterministic approach to this problem would be led astray and 
would be unable to find any desirable or undesirable incentives or dis- 
incentives for vertical integration. It is only by explicitly analyzing 
the effects of uncertainty on market behavior that the incentives and 
consequences of vertical integration can be fully comprehended. 



TJhite, op. cit. . p. 80 
^ibid.. p. 83 



44 



Appendix 1 



Let cost 2 = rK + p. ^ and cost 1 = rK + c. Notice that cost 2 

int 

represents the cost of producing one unit of the final good when the factor 

input is purchased from a stage 2 firm at price p. , while cost 1 is 

the resource cost when the factor input has been produced by a stage 1 

firm at the price c, where c is less than p. . Let Tr(i) stand for 

int 

the profits of a stage 1 firm when it holds sufficient input to satisfy 
i customers by itself. For the case of no vertical integration (i.e., 
i = 0) we have that 



•T-|(0) = [p- - cost 2] Z i pr(i) • (1 - X) -xCp) , where 
^ 

pr(i) = probability a firm obtains i customers, and 

x(p) = per capita demand. 

For the case where the stage 1 firm holds just enough of the input to 
satisfy one customer, we have 



Tr,(l) = [ I [[Pf - cost 1] + [p - cost 2](i - 1)(1 - A)] pr(i) - pr(0)-c] x(p) . 
i=l 

The expression for -n (0) is simply the net revenue per unit times the 
expected number of goods that are sold. The expression for w^ (1) is 
more complicated, and reflects the fact that if at least one customer 
appears, then the firm will be able to make a net profit on that customer 
of [p - cost 1], and a net profit of [p^ - cost 2] on each of the remaining 
customers. The term pr(0)«c reflects the risk that the firm will have 
spent c on production of the input, yet no customers will appear to use 
that input. Since Tr^(O) = 0, it follows that p. = cost 2, and that 



45 



TT (1) = [[Z pr(i)][p - cost 1] - pr(0)-c] x(p) , or 
1 



TT-. (1) = [[^ pr(i)][cost 2 - cost 1] - pr(0)'c] x(p), or 
1 

TTj^Cl) = [[I pr(i)][p^^j. - c] - pr(0).c] x(p) . (2; 

There will be an incentive for stage 1 firms to hold the input if 
IT (1) > TT,(0), or if 11^(1) > 0, or if 

[1 - Pr(0)] p.^^ > c , (3) 

where 1 - Pr(0) = the probability that at least one customer will frequent 
any stage 1 firm. 



46 



References 

K. Arrow - "Vertical Integration and Communication", Institute for Math- 
ematical Studies in the Social Sciences, October, 1974. 

K. Arrow - "Vertical Integration and Communication", The Bell Journal 
of Economics , Spring, 1975. 

D. Carlton - Market Behavior Under Uncertainty , unpublished Ph.D. thesis, 
M.I.T., September, 1975. 

A. Chandler, Jr. - Strategy and Structure; Chapters in the History of 
American Industrial Experience , M.I.T. Press, 1964. 

R. Coase - "The Nature of the Firm", Economics , November, 1937. 

J. Green - "Vertical Integration and Assurance of Markets", Discussion 
Paper 383, Harvard Institute of Economic Research, October, 1974. 

R. Hall - "An Aspect of the Economic Role of Unemployment", Proceedings 
of the International Economics Association , April, 1975. 

H. Malmgrem - "Information, Expectations, and the Theory of the Firm", 
Quarterly Journal of Economics , November, 1971. 

L. White - The Automobile Industry Since 1945 , Harvard University Press, 
Cambridge, Mass., 1971. 

0. Williamson - "The Vertical Integration of Production: Market Failure 
Considerations", American Economic Review , May, 1971. 



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